Explorations in Homeomorphic Variational Auto-Encoding

By Luca Falorsi, Pim de Haan, Tim R. Davidson, Nicola De Cao, Maurice Weiler, Patrick Forr'e, Taco Cohen. 2018

131 citations recorded by the source corpus.

The manifold hypothesis states that many kinds of high-dimensional data are concentrated near a low-dimensional manifold. If the topology of this data manifold is non-trivial, a continuous en-coder network cannot embed it in a one-to-one manner without creating holes of low density in the latent space. This is at odds with the Gaussian prior assumption typically made in Variational Auto-Encoders (VAEs), because the density of a Gaussian concentrates near a blob-like manifold. In this paper we investigate the use of manifold-valued latent variables. Specifically, we focus on the important case of continuously differen-tiable symmetry groups (Lie groups), such as the group of 3D rotations SO(3). We show how a VAE with SO(3)-valued latent variables can be constructed, by extending the reparameterization trick to compact connected Lie groups. Our exper-iments show that choosing manifold-valued latent variables that match the topology of the latent data manifold, is crucial to preserve the topological structure and learn a well-behaved latent space.

Subfields: Artificial Intelligence, Computational Mechanics, Computer Vision and Pattern Recognition, Generative Models

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Citations in the corpus

Stochastic Normalizing Flows (2020, background): “A major caveat of working with exactly invertible functions are topological constraints (Falorsi et al., 2018; 2019).”
Learning the Stein Discrepancy for Training and Evaluating Energy-Based Models without Sampling (2020, background): “…and sampling, but do so at the cost of a more restrictive model parameterization which can lead to well-known prob-Learning lems like posterior collapse in VAEs (Lucas et al., 2019), and the inability of NFs to model distributions with certain topological structures (Falorsi et al., 2018).”
Deferred neural rendering (2019, unknown): “For example, latent variables based on Lie groups can be used to better deal with object rotation [Falorsi et al. 2018].”
DeepVoxels: Learning Persistent 3D Feature Embeddings (2018, background): “Some of these approaches rely on embedding views into a latent space that does not enforce any geometrical constraints [54, 7, 9], others enforce geometric constraints in varying degrees [60, 45, 6, 10], such as learning rotationequivariant features by explicitly rotating the latent space feature ve”
The Power Spherical distribution (2020, background): “Examples include learning rotations ( i.e. , SO( n ) , Falorsi et al., 2018; 2019) and hierarchical structures on hyperbolic spaces (Mathieu et al., 2019; Nagano et al., 2019).”